The Elements of Coordinate Geometry

 
9781236159885: The Elements of Coordinate Geometry

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1896 Excerpt: ...m1m3 + m2(m1--m3) = m1m3-m22 (6). Also, (5) and (2) give 2m22 = (w + m3)2-2m1m3=m22-2m1m3, i.e. 7W22 + 2m1m3=0 (7). Solving (6) and (7), we have 2a-ft, „ rt 2a-h 7713=--=--, and 7M2 =-2 x. oa oa Substituting these values in (4), we have.2a--h h 2a-h I 2-„ oa a i.e. 21ak2 = 2(h-2a) so that the required locus is 21ay2=2(x-2af. 238. Ex. If the normals at three points P, Q, and R meet in a point O and S be the focus, prove that SP. SQ. SR = a. SO2. As in the previous question we know that the normals at the points (am12,-2am1), (am22,-2am2) and (am32,-2am3) meet in the point (ft, k) if m1 + m2 + m3 = 0 (1), 2a-ft m2m3 + m+711-2 =-----(2), k and m1m2m3=--(3). By Art. 202 we have 8P=al + ml2) SQ-a(l + m22)f and SR = al+m32). Hence SP'S®' SR = (l + m12) (l + m22) (l + m32) = 1 + (mx2 + m22 + ra32) + (m22m32 + m.m + mm?) + m-fm22m£. Also, from (1) and (2), we have mx2 + m22 + m32 = (m1 + m2 + ms)2-2 (?w2m3 + m3wi + wiw2) and m2%32 + mm-f + m12m22 = (m2mB + m37?i1 + ra)2-2m1m2m3 (mx + m2 + m3) = (-)',by(l)and(2). SP.SQ.SB, 07z-2a fh-2ay k2 Hence 1 = 1 + 2----+ ( +--a? a a J a _(h-a)2 + k2_S02 a2 a2 ' i.e. SP.SQ.SR = S02.a. EXAMPLES. XXX. Find the locus of a point 0 when the three normals drawn from it are such that 1. two of them make complementary angles with the axis. 2. two of them make angles with the axis the product of whose tangents is 2. 3. one bisects the angle between the other two. 4. two of them make equal angles with the given line y=mx + c. 5. the sum of the three angles made by them with the axis is constant. 6. the area of the triangle formed by their feet is constant. 7. the line joining the feet of two of them is always in a given direction. The normals at three points Pf Q, and 12 of the parabola y2--4ax meet in a point 0 who...

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Loney, S L
Editore: RareBooksClub.com (2012)
ISBN 10: 1236159888 ISBN 13: 9781236159885
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Descrizione libro RareBooksClub.com, 2012. Paperback. Condizione libro: New. This item is printed on demand. Codice libro della libreria INGM9781236159885

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2.

Loney, S L
Editore: RareBooksClub
ISBN 10: 1236159888 ISBN 13: 9781236159885
Nuovi Paperback Quantità: 20
Print on Demand
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BuySomeBooks
(Las Vegas, NV, U.S.A.)
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Descrizione libro RareBooksClub. Paperback. Condizione libro: New. This item is printed on demand. Paperback. 104 pages. Dimensions: 9.7in. x 7.4in. x 0.2in.This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1896 Excerpt: . . . m1m3 m2(m1--m3) m1m3-m22 (6). Also, (5) and (2) give 2m22 (w m3)2-2m1m3m22-2m1m3, i. e. 7W22 2m1m30 (7). Solving (6) and (7), we have 2a-ft, rt 2a-h 7713----, and 7M2 -2 x. oa oa Substituting these values in (4), we have. 2a--h h 2a-h I 2- oa a i. e. 21ak2 2(h-2a) so that the required locus is 21ay22(x-2af. 238. Ex. If the normals at three points P, Q, and R meet in a point O and S be the focus, prove that SP. SQ. SR a. SO2. As in the previous question we know that the normals at the points (am12, -2am1), (am22, -2am2) and (am32, -2am3) meet in the point (ft, k) if m1 m2 m3 0 (1), 2a-ft m2m3 m711-2 -----(2), k and m1m2m3--(3). By Art. 202 we have 8Pal ml2) SQ-a(l m22)f and SR alm32). Hence SPS SR (l m12) (l m22) (l m32) 1 (mx2 m22 ra32) (m22m32 m. m mm) m-fm22m. Also, from (1) and (2), we have mx2 m22 m32 (m1 m2 ms)2-2 (w2m3 m3wi wiw2) and m232 mm-f m12m22 (m2mB m37i1 ra)2-2m1m2m3 (mx m2 m3) (-), by(l)and(2). SP. SQ. SB, 07z-2a fh-2ay k2 Hence 1 1 2---- ( --a a a J a (h-a)2 k2S02 a2 a2 i. e. SP. SQ. SR S02. a. EXAMPLES. XXX. Find the locus of a point 0 when the three normals drawn from it are such that 1. two of them make complementary angles with the axis. 2. two of them make angles with the axis the product of whose tangents is 2. 3. one bisects the angle between the other two. 4. two of them make equal angles with the given line ymx c. 5. the sum of the three angles made by them with the axis is constant. 6. the area of the triangle formed by their feet is constant. 7. the line joining the feet of two of them is always in a given direction. The normals at three points Pf Q, and 12 of the parabola y2--4ax meet in a point 0 who. . . This item ships from La Vergne,TN. Paperback. Codice libro della libreria 9781236159885

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